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. 2019 Oct 30;9:15607. doi: 10.1038/s41598-019-51575-1

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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(a,b) Cancer cell S and D evolution near fixed points $ℱ_{1}$ and $ℱ_{2}$ . (c,d) Streamlines determine the direction of attraction or repulsion near the fixed points for the tumor cells and the chemicals. Due to the high dimensionality of the system we project the dynamics along two directions, while keeping the other two constant. In (c,d) A complex pair of eigenvalues with negative real part determines a spiral point in the {S, D} plane. Differently, the other two eigenvalues are real and negative, making the point $ℱ_{2}$ globally stable (c). (c,d) arrows determine the direction of attraction or repulsion near the fixed points for the tumor cells and the chemicals and the colors, increasing from purple to red, are the relative norms of the vector fields. Due to the high dimensionality of the system, we project the dynamics along two directions, while keeping the other two constant.